Question

Why is measuring the duration of a number of swings a better way to determine the period of a pendulum than by measuring a single cycle?

Answer 1

Answer:

It is to reduce the expected relative error of the measurement.

Explanation:

If there was a way to measure without error, this method would be unnecessary. In practice, the pesky error is always there. The sources are varied: inexact instrument, small inaccuracies in starting/stopping the timer, etc. But, it is reasonable to assume that such an error is random and has an expected spread that is independent of the actual duration of measurement. Under such assumptions, the methods offers a great advantage:

Let ε denote an additive measurement error. Let the error be random, symmetric (negative/positive), distributed in some fixed range independent of the actual measured value. The error represents an additive component in our measurement, i.e., (measurement) = (true value) + (error). In the case of one period T, we get to measure the duration T':

$T' = T + \epsilon$

so the relative error is

$\frac{|T'-T|}{T}=\frac{|T+\epsilon-T|}{T}=\frac{|\epsilon|}{T}$

In a separate experiment, suppose you measure n periods. Same error applies:

$T_n'=n\cdot T+\epsilon$

we can get a single period by dividing the measured value by n:

$\frac{T_n'}{n}=\frac{n\cdot T +\epsilon}{n}=T+\frac{\epsilon}{n}$

and the relative error of such a result will be:

$\frac{|T+\frac{\epsilon}{n}-T|}{T}=\frac{|\epsilon|}{n\cdot T}$

which is n times smaller than the relative error of the single measurement above. The more periods are included in the measurement, the smaller the expected error!